Large prefixes on a fixed finite prime support

A divisor-tuple majorant covers the third class. Its proof uses a single smooth divisor of the radical value and does not remove any bad-prime powers.

def NairTenenbaum.PolynomialSystem.HasLargeSmoothDivisorTuple {k r : ℕ} (P : PolynomialSystem k r) (s : Finset ℕ) (ε x : ℝ) (t : ℤ) : Prop

A polynomial value admits a smooth divisor tuple above the lower mass cutoff.

Equations

• One or more equations did not get rendered due to their size.

theorem NairTenenbaum.largeDivisor_sieve_level {g : ℕ} {ε x y : ℝ} (hx : 0 < x) (hgap : 2 ≤ x ^ (↑g ^ 2 * ε)) (hy : x ^ (4 * ↑g ^ 2 * ε) ≤ y) : x ^ (3 * ↑g ^ 2 * ε) ≤ y / 2

The large-divisor upper cutoff is below half the available interval.